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We investigate symmetry properties of peculiar modules, a Heegaard Floerinvariant of 4-ended tangles which the author introduced in [arXiv:1712.05050].In particular, we give an almost complete answer to the geography problem forcomponents of peculiar modules of tangles. As a main application, we show thatConway mutation preserves the hat flavour of the relatively $\delta$-gradedHeegaard Floer theory of links.

On symmetries of peculiar modules; or, $\delta$-graded link Floer homology is mutation invariant